A152187 a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=1, a(1)=5.

1, 5, 20, 85, 355, 1490, 6245, 26185, 109780, 460265, 1929695, 8090410, 33919705, 142211165, 596232020, 2499751885, 10480415755, 43940006690, 184222098845, 772366329985, 3238209484180, 13576460102465, 56920427728295

Offset

0,2

Comments

Unsigned version of A152185.

From Johannes W. Meijer, Aug 01 2010: (Start)

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given side square (m = 2, 4, 6 and 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.

The sequence above corresponds to 24 red king vectors, i.e., A[5] vectors, with decimal values 27, 30, 51, 54, 57, 60, 90, 114, 120, 147, 150, 153, 156, 177, 180, 210, 216, 240, 282, 306, 312, 402, 408 and 432. These vectors lead for the corner squares to A015523 and for the central square to A179606.

This sequence belongs to a family of sequences with g.f. (1+2*x)/(1 - 3*x - k*x^2). Red king sequences that are members of this family are A007483 (k=2), A108981 (k=4), A152187 (k=5; this sequence), A154964 (k=6), A179602 (k=7) and A179598 (k=8). We observe that there is no red king sequence for k=3. Other members of this family are A036563 (k=-2), A054486 (k=-1), A084244 (k=0), A108300 (k=1) and A000351 (k=10).

Inverse binomial transform of A015449 (without the first leading 1).

(End)

Links

Table of n, a(n) for n=0..22.

Index entries for linear recurrences with constant coefficients, signature (3, 5).

Formula

G.f.: (1+2*x)/(1 - 3*x - 5*x^2).

Lim_{k->infinity} a(n+k)/a(k) = (A072263(n) + A015523(n)*sqrt(29))/2. - Johannes W. Meijer, Aug 01 2010

G.f.: G(0)*(1+2*x)/(2-3*x), where G(k) = 1 + 1/(1 - x*(29*k-9)/(x*(29*k+20) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013

Mathematica

LinearRecurrence[{3, 5}, {1, 5}, 40] (* Harvey P. Dale, May 03 2013 *)

Crossrefs

Cf. A015523, A072263, A072264, A179606, A197189.

Sequence in context: A069007 A126987 A152185 * A341920 A045499 A358518

Adjacent sequences: A152184 A152185 A152186 * A152188 A152189 A152190

Keyword

nonn,easy

Author

Philippe Deléham, Nov 28 2008

Status

approved